A307650 Number of regions in the Shi-threshold arrangement STn.

1, 1, 3, 27, 345, 5513, 106619, 2426819, 63595653, 1886225589, 62471885847, 2285488346111, 91536241567505, 3983627374435361, 187189099298640435, 9445664642509417563, 509426043718487709501, 29243389264220375989229, 1780261345776857866401263, 114560896953217093009807895

Offset

0,3

Links

Table of n, a(n) for n=0..19.

A. R. Balasubramanian, Generalized threshold arrangements, arXiv:1904.08903 [math.CO], 2019. Says that sequence is 1, 27, 345, 5513, ... for n>=2.

Seunghyun Seo, Shi threshold arrangement, Electronic Journal of Combinatorics, volume 19, issue 3, 2012.

Formula

a(n) = Sum_{j>=0} binomial(2*j+1, j)*j!*(-1)^(n-j)*Stirling2(n,j) + 2*n*Sum_{j>=0} binomial(2*j+2, j)*j!*(-1)^(n-j)*Stirling2(n-1,j) + n*(n-1)*Sum_{j>=0} binomial(2*j+3, j)*j!*(-1)^(n-j)*Stirling2(n-2,j).

E.g.f.: C(1-exp(-x))*(x*C(1-exp(-x))-1)^2/sqrt(4*exp(-x)-3) where C(z) = (1-sqrt(1-4*z))/(2*z) is the g.f. for Catalan numbers.

Prog

(PARI) a(n) = sum(j=0, n, binomial(2*j+1, j)*j!*(-1)^(n-j)*stirling(n, j, 2)) + 2*n*sum(j=0, n-1, binomial(2*j+2, j)*j!*(-1)^(n-j)*stirling(n-1, j, 2)) + n*(n-1)*sum(j=0, n-2, binomial(2*j+3, j)*j!*(-1)^(n-j)*stirling(n-2, j, 2));

Crossrefs

Cf. A000108 (Catalan numbers), A008277 (Stirling2).

Sequence in context: A153853 A067000 A354658 * A168593 A364431 A328182

Adjacent sequences: A307647 A307648 A307649 * A307651 A307652 A307653

Keyword

nonn

Author

Michel Marcus, Apr 20 2019

Status

approved